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Existence Theory for Nonlinear Ordinary Differential Equations

  • Book
  • © 1997

Overview

Authors:
  1. Donal O’Regan

    1. Department of Mathematics, University College Galway, Galway, Ireland

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Part of the book series: Mathematics and Its Applications (MAIA, volume 398)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-vii
    Download chapter PDF

  2. Introduction, notation and preliminaries

    • Donal O’Regan
    Pages 1-3

  3. Fixed point theory

    • Donal O’Regan
    Pages 4-10

  4. Initial value problems

    • Donal O’Regan
    Pages 11-21

  5. First order periodic problems

    • Donal O’Regan
    Pages 22-33

  6. Existence principles for second order boundary value problems

    • Donal O’Regan
    Pages 34-45

  7. Boundary value problems without growth restrictions

    • Donal O’Regan
    Pages 46-57

  8. Positone boundary value problems

    • Donal O’Regan
    Pages 58-66

  9. Semi-positone boundary value problems

    • Donal O’Regan
    Pages 67-81

  10. Differential equations singular in the solution variable

    • Donal O’Regan
    Pages 82-104

  11. Existence principle for singular boundary value problems

    • Donal O’Regan
    Pages 105-115

  12. Nonresonance problems in the limit circle case

    • Donal O’Regan
    Pages 116-132

  13. Resonance problems in the limit circle case

    • Donal O’Regan
    Pages 133-155

  14. Boundary value problems on the half line

    • Donal O’Regan
    Pages 156-163

  15. Existence theory for ordinary differential equations on compact and noncompact intervals

    • Donal O’Regan
    Pages 164-173

  16. Impulsive differential equations

    • Donal O’Regan
    Pages 174-185

  17. Differential equations in abstract spaces

    • Donal O’Regan
    Pages 186-194

  18. Back Matter

    Pages 195-200
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Keywords

About this book

We begin our applications of fixed point methods with existence of solutions to certain first order initial initial value problems. This problem is relatively easy to treat, illustrates important methods, and in the end will carry us a good deal further than may first meet the eye. Thus, we seek solutions to Y'. = I(t,y) (1. 1 ) { yeO) = r n where I: I X R n ---+ R and I = [0, b]. We shall seek solutions that are de­ fined either locally or globally on I, according to the assumptions imposed on I. Notice that (1. 1) is a system of first order equations because I takes its values in Rn. In section 3. 2 we will first establish some basic existence theorems which guarantee that a solution to (1. 1) exists for t > 0 and near zero. Familiar examples show that the interval of existence can be arbi­ trarily short, depending on the initial value r and the nonlinear behaviour of I. As a result we will also examine in section 3. 2 the dependence of the interval of existence on I and r. We mention in passing that, in the results which follow, the interval I can be replaced by any bounded interval and the initial value can be specified at any point in I. The reasoning needed to cover this slightly more general situation requires minor modifications on the arguments given here.

Reviews

`I warmly recommend it to everyone interested in ordinary differential equations and its applications.'
Acta Sci. Math., 64 (1998)

Authors and Affiliations

  • Department of Mathematics, University College Galway, Galway, Ireland

    Donal O’Regan

Bibliographic Information

  • Book Title: Existence Theory for Nonlinear Ordinary Differential Equations

  • Authors: Donal O’Regan

  • Series Title: Mathematics and Its Applications

  • DOI: https://doi.org/10.1007/978-94-017-1517-1

  • Publisher: Springer Dordrecht

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media Dordrecht 1997

  • Hardcover ISBN: 978-0-7923-4511-4Published: 30 April 1997

  • Softcover ISBN: 978-90-481-4835-6Published: 06 December 2010

  • eBook ISBN: 978-94-017-1517-1Published: 17 April 2013

  • Edition Number: 1

  • Number of Pages: VII, 200

  • Topics: Ordinary Differential Equations, Integral Equations, Operator Theory

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